y = x 2 − 2 x y = x + 4 {\displaystyle y=x^{2}-2x\qquad y=x+4} x = 4 x = − 1 {\displaystyle x=4\qquad x=-1} ∫ − 1 4 ( x + 4 − x 2 + 2 x ) d x {\displaystyle \int _{-1}^{4}(x+4-x^{2}+2x)dx} = | − 1 4 3 2 x 2 + 4 x − 1 3 x 3 {\displaystyle ={\Bigg |}_{-1}^{4}{\frac {3}{2}}x^{2}+4x-{\frac {1}{3}}x^{3}} = 3 2 ( 4 ) 2 − 1 3 ( 4 ) 3 + 3 2 ( − 1 ) 2 + 4 ( − 1 ) − 1 3 ( − 1 ) 3 {\displaystyle ={\frac {3}{2}}(4)^{2}-{\frac {1}{3}}(4)^{3}+{\frac {3}{2}}(-1)^{2}+4(-1)-{\frac {1}{3}}(-1)^{3}} = 125 6 {\displaystyle ={\frac {125}{6}}}
